Critical behavior within 20 fs drives the out-of-equilibrium laser-induced magnetic phase transition in nickel

Abstract

It has long been known that ferromagnets undergo a phase transition from ferromagnetic to paramagnetic at the Curie temperature, associated with critical phenomena such as a divergence in the heat capacity. A ferromagnet can also be transiently demagnetized by heating it with an ultrafast laser pulse. However, to date, the connection between out-of-equilibrium and equilibrium phase transitions, or how fast the out-of-equilibrium phase transitions can proceed, was not known. By combining time- and angle-resolved photoemission with time-resolved transverse magneto-optical Kerr spectroscopies, we show that the same critical behavior also governs the ultrafast magnetic phase transition in nickel. This is evidenced by several observations. First, we observe a divergence of the transient heat capacity of the electron spin system preceding material demagnetization. Second, when the electron temperature is transiently driven above the Curie temperature, we observe an extremely rapid change in the material response: The spin system absorbs sufficient energy within the first 20 fs to subsequently proceed through the phase transition, whereas demagnetization and the collapse of the exchange splitting occur on much longer, fluence-independent time scales of ~176 fs. Third, we find that the transient electron temperature alone dictates the magnetic response. Our results are important because they connect the out-of-equilibrium material behavior to the strongly coupled equilibrium behavior and uncover a new time scale in the process of ultrafast demagnetization.

Fig. 1. Schematic of the critical behavior of ultrafast demagnetization in Ni.

(A) After excitation by a femtosecond laser pulse above the critical fluence (F_c), the transient electron temperature (T_e) is driven above the Curie temperature (T_c), inducing high-energy spin excitations within 20 fs, which store the magnetic energy (see text). The Fermi-Dirac distributions of electrons are also plotted. Demagnetization occurs later, in ~176 fs, driven by relaxation of nonequilibrium spins and the likely excitation of low-energy magnons. Full recovery of the spin system occurs within ~500 fs to ~76 ps, depending on the laser fluence. (B and C) Experimental setups for time-resolved ARPES and TMOKE, respectively, using ultrafast high-harmonic sources. IR, infrared.

Fig. 2. Magnetization dynamics in Ni.

(A) Change of the TMOKE asymmetry and exchange splitting reduction ΔE_ex as a function of time delay for different laser fluences. The solid lines represent fitting results, from which we extract the three characteristic times for demagnetization (τ_demag), fast recovery (τ_recover1), and slow recovery (τ_recover2) (see Materials and Methods). The fit to TMOKE (upper panel) and ARPES (lower panel and lower fluence) yields the same fluence-independent time constants. a.u., arbitrary units. (B) Typical TMOKE asymmetry before (t_d = −360 fs) and after (t_d = 500 fs) excitation with a pump fluence F ≈ 6 mJ/cm². (C) Photoelectron spectra of Ni(111) along the $\overline{\mathrm{\Gamma }}-\overline{\mathit{K}}$ direction before (t_d = −500 fs) and after (t_d = 500 fs) laser excitation, showing the collapse in the exchange splitting E_ex after excitation (blue dashed lines). The dashed-dotted lines represent the momentum at which photoemission intensities are extracted. The photoemission intensities are plotted in the right panel with E_ex extracted from a Voigt function fit to the data (dashed lines; see the Supplementary Materials). (D) Constant, fluence-independent demagnetization time observed for different laser fluences for both ARPES and TMOKE.

Fig. 3. Ultrafast charge dynamics in Ni.

(A) Log plots of the photoemission intensity above E_F for F ≈ 6 mJ/cm² and at different t_d, integrated from k_// ≈ 0.85 Å⁻¹ to k_// ≈ 1.3 Å⁻¹ in the momentum space. The dashed lines represent the fitting of the photoemission intensities with the Fermi-Dirac distribution convolved with experimental energy resolution (see the Supplementary Materials). Inset: Integrated photoemission intensity as a function of pump-probe time delay. The yellow dashed box illustrates the integration region of electron population in (B). (B) Dynamics of the electron temperature and the relative electron population (n/n₀) within ~0.2 eV above E_F as a function of t_d. The electron population is normalized to the band electron population (n₀) ~0.2 eV below E_F (see the Supplementary Materials). (C) Comparison of the electron temperature [red dashed line, same as (B)] and the change of EUV transient reflectivity at a similar pump fluence. Inset: EUV transient reflectivity measurement. The resonant EUV light (65 eV) directly probes the charge dynamics around E_F induced by the laser pump pulse. This measurement is averaged over k-space.

Fig. 4. Observation of multiple critical behaviors during ultrafast demagnetization in Ni.

(A) Peak electron temperature extracted ~24 fs after excitation as a function of pump fluence. The open symbols represent the electron temperature extracted at different k_// using Tr-ARPES. The solid red line is the fit using Eq. 1 considering the transient electron and magnetic heat capacity [inset of (B)], whereas the green dashed line considers only the contribution from transient electron heat capacity (see the Supplementary Materials). The yellow-colored region (ΔF_S) is the energy transferred to the spin system within ~20 fs. (B) Change in the exchange splitting at 2 ps as a function of pump fluence. The red line represents a fit with an error function. The same critical fluence of F_c ≈ 2.8 mJ/cm² is observed for the exchange splitting collapse and the peak electron temperature in (A). The transient electron heat capacity is plotted in the inset. (C) Peak electron temperature calculated using Eq. 1 and (C_e + C_m) Transient [inset of (B)] for the sample temperatures of 300 and 100 K. The red solid line is the same as in (A). (D) Change of exchange splitting at 2 ps as a function of laser fluence at different sample temperatures. The solid lines represent the error function fit of the experimental results. The dashed lines align the critical fluences observed in (C) and (D) for different sample temperatures.